The compressor is a component of the air conditioning system that is responsible for compressing the refrigerant gas and circulating it through the system. It is not directly affected by the furnace or the thermostat, and will continue to work normally as long as the air conditioning system is properly maintained.
The furnace and the thermostat are responsible for regulating the temperature of the air inside the home. The furnace heats the air and the thermostat controls the temperature by turning the furnace on and off as needed. The air conditioner compressor is responsible for cooling the air and is controlled by a separate thermostat.
Replacing the furnace and the thermostat will not improve the performance of the air conditioner compressor or the air conditioning system as a whole. If the air conditioning system is not working properly, it is recommended to have a qualified technician diagnose the problem and perform any necessary repairs or maintenance.